NEW YORK (AP) -- Sears' biggest shareholder has suggested the company sell its well-known Kenmore brand and some real estate holdings, offering itself as a potential buyer.

The ailing company has sold off other major brands as it struggles to stay afloat, with Kenmore a notable remainder of the powerhouse retailer that survived two world wars and the Great Depression.

The private equity firm ESL Investment, headed by Sears chairman and CEO Edward Lampert, said it might buy the assets — Kenmore, Sears Home Services' home improvement business and the company's Parts Direct business — if the company is willing to sell.

That sent shares of Sears Holding Corp., which have lost more than 70 percent of their value in the past year, up nearly 5 percent.

Lampert, who combined Sears and Kmart in 2005 after helping bring Kmart out of bankruptcy, has long pledged to turn the company's fortunes around. He said the retailer would find ways to capitalize on its best-known brands like Kenmore appliances and DieHard car batteries, as well as its vast holdings of land.

But the company has continued to see shoppers move on to Target, Walmart and Amazon, and has closed hundreds of stores, cut costs and sold brands to deal with falling sales.

In his letter to the board, Lampert said Sears has been trying to sell the Kenmore businesses for nearly two years but it has been unable to do so.

Kenmore could have substantial value. Amazon.com began selling Kenmore appliances on its site almost a year ago. ESL has not placed a potential value on Kenmore, but said its non-binding proposal gives the services and home improvement units an enterprise value of $500 million.

ESL said it also would be open to making an offer for Sears' real estate, including the assumption of $1.2 billion in debt.

"In our view, pursuing these divestures now will demonstrate the value of Sears' portfolio of assets, will provide an important source of liquidity to Sears and could avoid any deterioration in the value of such assets," Lampert wrote.

Sears, which started in the 1880s as a mail-order catalog business, was a back-to-school and appliance shopping destination for generations. Its storied catalog featured items from bicycles to sewing machines to houses, and the company's stores were a fixture of suburban malls from the 1950s to 1970s.